Differential Equations

Lecture 24 & page 412

Introduction

A differential equation is defined as any equation that involves an unknown function and at least one of its derivatives.

The order of a differential equation is the order of the highest derivative that occurs in that equation.

A function is called a solution of a differential equation if the equation is satisfied when $y = f(x)$ and its derivatives are substituted into the equation. in other words, a solution to $y' = xy$ would need to be in the form:

$f'(x) = xf(x)$

Separable equations

A separable equation is a first order differential equation that can be written in the form:

$\dfrac{dy}{dx} = g(x)f(y)$

The term separable comes from the idea that the function of $x$ and the function of $x$ could be separated on to different sides of the equals. This would look something like this:

$h(y) \\, dy = g(x) \\, dx$

General & particular

Logistic growth